Exercise 6, § 1

a) Let $I = [0, +\infty[$; if $f$ is Lipschitz on $I \times E$ for a regulated function $k(t) > 0$ such that the integral $\int_0^{+\infty} k(t) dt$ converges, show that every integral of the equation $x' = f(t, x)$ is defined on all of $I$.

b) If further the integral $\int_0^{+\infty} \|f(t, x_0)\| dt$ converges (for a certain point $x_0 \in E$), show that every integral of $x' = f(t, x)$ tends to a finite limit as $t$ tends to $+\infty$ (first show that every integral is bounded as $t$ tends to $+\infty$).